On symmetry breaking of dual polyhedra of non-crystallographic group H3

The study of the polyhedra described in this paper is relevant to the icosahedral symmetry in the assembly of various spherical molecules, biomolecules and viruses. A symmetry-breaking mechanism is applied to the family of polytopes V-H3 (lambda) constructed for each type of dominant point lambda. Here a polytope V-H3 (lambda) is considered as a dual of a D-H3 (lambda) polytope obtained from the action of the Coxeter group H-3 on a single point lambda is an element of R-3. The H-3 symmetry is reduced to the symmetry of its two-dimensional subgroups H-2, A(1) x A(1) and A(2) that are used to examine the geometric structure of V-H3 (lambda) polytopes. The latter is presented as a stack of parallel circular/polygonal orbits known as the 'pancake' structure of a polytope. Inserting more orbits into an orbit decomposition results in the extension of the V-H3 (lambda) structure into various nanotubes. Moreover, since a V-H3 (lambda) polytope may contain the orbits obtained by the action of H-3 on the seed points (a, 0, 0), (0, b, 0) and (0, 0, c) within its structure, the stellations of flat-faced V-H3 (lambda) polytopes are constructed whenever the radii of such orbits are appropriately scaled. Finally, since the fullerene C-20 has the dodecahedral structure of V-H3 (a, 0, 0), the construction of the smallest fullerenes C-24, C-26, C-28, C-30 together with the nanotubes C20+6N, C20+ 10N is presented.